
where \(j^{\prime }\)  is given by (REF ). We note that with the above discussion, it can be easily seen that directly solving (REF ) is extremely difficult. Due to the interdependence of the leader's commitment and the follower's best response, the objective function is not linear nor convex and not even a continuous function with respect to \(\mathbf {x}, \mathbf {C}\) . Therefore, directly solving (REF ) as in previous works [1]}, [2]}, [3]} is not viable in the proposed sigBSG, and we need another approach to tackling this problem. Inspired by the probabilistic viewpoint adopted in Bayesian persuasion [4]}, [5]}, in the following, we rewrite the signaling commitment as a probability measure, which greatly simplifies the problem.
